1. Field
The present invention relates to the field of microelectromechanical systems (MEMS), and more particularly to temperature compensated MEMS resonators.
2. Discussion of Related Art
“MEMS” generally refers to apparatus incorporating some mechanical structure having a dimensional scale that is comparable to microelectronic devices. For example, less than approximately 250 um. This mechanical structure is typically capable of some form of mechanical motion and is formed at the micro-scale using fabrication techniques similar to those utilized in the microelectronic industry such as thin film deposition, and thin film patterning by photolithography and reactive ion etching (RIE). The micromechanical structure in a MEMS distinguishes a MEMS from a microelectronic device.
Certain MEMS include a resonator. MEMS resonators are of particular interest in timing devices for an integrated circuit (IC). The resonator may have a variety of physical shapes, such as, but not limited to, beams and plates. Beams may be anchored on two ends or just one. A beam anchored at only one end is frequently referred to as a cantilevered beam. MEMS 100, employing a conventional beam resonator, is shown in FIG. 1A. MEMS 100 includes over substrate 101, drive/sense electrodes 150 and a beam resonator 105. A cross-sectional view along the line a-a′ of beam resonator 105 depicted in Figure 1A is shown in FIG. 1B. As the cross-section view shows, beam resonator 105 comprises a single material 130.
A resonator has resonant modes (e.g. flexural, bulk, etc.) of particular frequencies that depend at least upon the physical shape, size and stiffness of the material employed for the resonator. The stiffness of a material, characterized as Young's modulus, is generally temperature dependent.
For a MEMS resonator, such as beam resonator 105, comprising a single material and therefore having uniform density and mechanical properties, the frequency of all modes and shapes can be derived to be a function of the material Young's modulus, E , the density, ρ, and a dimensionless constant , κ, or:
                    f        =                              κ                          Λ              ⁡                              (                                  d                  i                                )                                              ⁢                                    E              ρ                                                          (                  Equation          ⁢                                          ⁢          1                )            
In Equation 1, Λ(di) is a function of the geometric dimensions diof the resonator and has units of length.
The temperature dependence of the resonator frequency is independent of the form of Λ(di) assuming a linear temperature dependence for these quantities of the form:E(T)=E0(1+γ(T−T0)  (Equation 2)di(T)=di0(1+α(T−T0))  (Equation 3)ρ(T)=ρ0(1−3α(T−T0))  (Equation 4)so that:
                                                                        1                E                            ⁢                                                ∂                  E                                                  ∂                  T                                                      ⁢                          ❘                              T                =                0                                              =          γ                ,                                                            1                                  d                  i                                            ⁢                                                ∂                                      d                    i                                                                    ∂                  T                                                      ⁢                          ❘                              T                =                0                                              =          α                ,                                                            1                ρ                            ⁢                                                ∂                  ρ                                                  ∂                  T                                                      ⁢                          ❘                              T                =                0                                              =                                    -              3                        ⁢                                                  ⁢                          α              .                                                          (                  Equation          ⁢                                          ⁢          5                )            
The temperature dependence of resonator frequency may then be expressed in terms of the linear coefficient of thermal expansion (CTE), α, and the Young's modulus temperature coefficient, γ:
                              f          ⁡                      (            T            )                          =                              f            0                    ⁡                      (                          1              +                                                1                  2                                ⁢                                  (                                      γ                    +                    α                                    )                                ⁢                                  (                                      T                    -                                          T                      0                                                        )                                                      )                                              (                  Equation          ⁢                                          ⁢          6                )            
Typical resonators, comprising semiconductor materials, such as single crystalline or polycrystalline silicon, have a Young's modulus that decreases with temperature. Thus, resonators comprising such a resonator will generally have a resonant frequency that decreases with increasing temperature. For an exemplary polycrystalline silicon-germanium (SiGe) resonator, the experimentally determined values are γ=−1.075×10−4/° C. and α=4.52×10−6/° C. Because the magnitude of γ is approximately 20 times larger than that of α, the temperature coefficient of frequency (TCF) is negative for a homogeneous SiGe resonator of any shape and in any mode. Using the values above, the TCF is approximately −51.49×10−6/° C. or −51.49 ppm/° C.
Due in part to the temperature dependence of the Young's modulus, fabricating MEMS resonators having temperature sensitivities on the same order of magnitude as existing quartz resonators is therefore challenging. For example, quartz, being relatively temperature stable, has a frequency drift of approximately 0.5 parts per million (ppm) per degree Celsius (° C.), while conventional MEMS resonators consisting of homogeneous materials of uniform density and mechanical properties have drifts on the order of 100 times higher, or 50 ppm/° C. Thus, widespread adoption of MEMS resonators in IC timing devices may require compensating temperature induced frequency variation.